How can I process very large darkroom prints when I don't have trays that size?

Yes. A common low-space method is to process the print flat on a waterproof surface rather than in oversized trays. Lay the exposed print on a plastic drop cloth or other waterproof work surface under safelight, then apply developer, stop bath, and fixer with a large soft brush, paint roller, or sponge. Use gloves, keep the surface evenly wet, and expect development times to be somewhat longer than with full immersion.

Another practical option is to create more usable wet-side space by bridging your sink with sealed boards and setting trays on top, so you can drain chemicals into the sink without handling full trays awkwardly. A simple homemade waterproof trough or larger sink can also work.

Whichever method you use, make small test prints first to minimize waste and dial in exposure and processing. After fixing, wash the print where you have enough room and water flow; for very large prints, people sometimes wash them outside with a hose.

So the proven approaches are: process flat with brushed-on chemistry, or expand your wet workspace enough to support large trays or a trough.